“This one blew our minds.”
Linda Jaslow has been working on fractions in a first-grade classroom using equal sharing problems. She sent this description of one girl’s insight about how to generate equivalent fractions:
We also had an amazing revelation by a first grader on equivalent fractions. The class had done a great deal of playing with the fractions and were pretty able in terms of equivalence. But this first grader blew our minds. The problem was 8 [people] sharing 14 [of something, e.g., pancakes]. So we had the different responses but one little girl said that 14/8 was equivalent to 7/4 because 2/8 was equal to 1/4 so you could double the 7/4 and get 14/8 or you could half 14/8 to get 7/4. It was too close to 7/4 x 2/2 idea to be an accident…. I am not sure how far you would take this with a first grader but it felt like a teachable moment to explore. Curious…would you have just let that big idea slide or done something with it?
I wonder how she arrived at that conclusion. I think she might be reasoning by analogy — a form of reasoning not specific to mathematics — rather than from mathematical principles — such as multiplication by 1 = 2/2. I’m also curious whether she can relate her ideas back to the sharing situation (e.g., a child in a group of 8 people sharing 14 pancakes gets as much pancake as a child in a group of 4 people sharing 7 pancakes) and to what kinds of other fraction relationships she can extend this reasoning. My guess is it would be harder for her to reason about equivalence relationships that involved tripling or “thirding” the numerators and denominators, and so on. But all of this doesn’t make her insight any less fascinating or powerful in the context of learning fractions. And yes, I’d definitely want to explore it!
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